JOURNAL OF THE OPTICAL SOCIETY OF AMERICA Mie Scattering Functions for Refractive Index of 2.105* MILTONKERKER AND EGON MATIJEVIf Department of Chemistry, Clarkson Collegeof Technology, Potsdam, New York (Received July 8, 1960) Values of intensity functions and total scattering coefficient are presented for n=2.105 and a=0. 2 (0.4) 5.8; 6.0 (0.2) 15.0. The applicability of Penndorf's approximate methods to the total scattering coefficient is discussed. WE have computed Mien scattering functions for '~v refractive index of 2.105 in connection with a light-scattering study of silver chloride aerosols. In addition to their usefulness in studying this particular experimental system, these computations make avail- able a large number of results for a refractive index in a range where there have been very few data. This will now permit a more careful analysis of the influence of refractive index upon the scattering functions and also the testing of existing interpolation procedures as well as the development of improved ones. The results are given in Table I. The size parameter a is equal to 2 rr/X, where r is the particle radius and X the wavelength of radiation in the medium. The angle of observation y is the angle between the scattering direction and the reversed direction of the incident beam; hence 0° is the backward direction. The angular distribution functions or intensity func- tions, i and i 2 , are proportional to the intensity of the perpendicular and horizontal components of the scattered light. For detailed definitions, the reader is referred to the standard literature.- The total scattering coefficientK is the total flux scattered by a particle in all directions relative to the flux incident on the geometrical cross section of the particle. The digit following each number is the decimal exponent, e.g., 1.8099 -5 means 1.8099X10-1. All computations were carried out with aid of the IBM 704 computer. For values of a less than 6, the Legendre functions were read into the computer from the tabulations of Gumprecht and Sliepcevich 7 and, in the cases for -y=450 and 1350, from our previous *Supported in part by contract with the U. S. Army Chemical Corps. I G. Mie, Ann. Physik 25, 377 (1908). 2 E. Matijevi6, M. Kerker, and K. F. Schulz, Discussions Faraday Soc. (September, 1960). 3 R. 0. Gumprecht and C. M. Sliepcevich, Light Scattering Functions for Spherical Particles (University of Michigan Press, Ann Arbor, Michigan, 1951). 4 W. J. Pangonis, W. Heller, and A. Jacobson, Tables of Light Scattering Functions for Spherical Particles (Wayne State Univer- sity Press, Detroit, Michigan, 1957). 5A. N. Lowan, Tables of Scattering Functions for Spherical Particles, Natl. Bur. Standards, (U. S.), Appl. Math. Series 4, Washington, Government Printing Office (1948). 6 H. C. Van de Hulst, Light Scattering l1aSmall Particles (John Wiley & Sons, Inc., New York, 1957). R K. 0. Gumprecht and C. M. Sliepcevich, Functions of Partial Derivatives of Legendre Polynomials (University of Michigan Press, Ann Arbor, Michigan, 1951). calculations.' The necessary Bessel functions were computed internally with the aid of the usual recursion relations. 9 This Bessel function routine failed for larger values of a (a> 6) when higher-order functions were called for because of the loss of accuracy upon con- tinued iteration of the recursion relation. For a> 6, a program developed by the National Bureau of Standards was utilized. In order to maintain the necessary precision in the calculation of the Bessel functions, a double-precision interpretive system was used (20 decimal places). In this case the Legendre functions were also computed internally using recursion relations. Despite the interest in scattering phenomena in such diverse areas as chemistry, meteorology, astrophysics, space science, microwave physics, etc., the number of available scattering functions is still much too small to cover the whole range of useful applications. All these functions contain the two parameters, n and a. Van de Hulst 0 has developed a lucid presentation of scattering functions in terms of an m-a plane such that to each point in the plane there exists a particular set of functions corresponding to a particular scattering system. The problem has been that not only is the region of the m- a plane where exact Mie theory computations are required quite extensive, but because of the fluc- tuating character of the functions, an extremely high density of points is necessary. This, in turn, has led to a need for accurate interpolation schemes in order to obtain values for points in the plane intermediate between those available by exact computation. Penndorf"' 12 has recently collected together and studied the existing values of the total scattering coefficient K for real values of the refractive index. His purpose was to develop numerical formulas which would permit rapid calculation of approximate values of K for intermediate refractive indexes. Since a large number of computations were available for refractive 8 M. Kerker and E. Matijevi6, J. Opt. Soc. Am. 50, 722 (1960). 9 G. N. Watson, Theory of Bessel Functions (Cambridge Univer- sity Press, Cambridge, Massachusetts, 1948). 10 H. C. Van de Hulst, Optics of Spherical Particles (Recherches Astronomiques de l'Observatoire d'Utrecht, XI part 1, Amsterdam, 1946). 11 R. Penndorf, New Tables of Mie Scattering Functions for Spherical Particles (Geophysical Research Papers No. 45, Part 6, Air Force Cambridge Research Center, Bedford, Massachusetts, March, 1956). 12 R. Penndorf, J. Phys. Chem. 62, 1537 (1958). 87 VOLUME 51, NUMBER JANUARY, 1961
Transcript
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
Mie Scattering Functions for Refractive Index of 2.105*
MILTON KERKER AND EGON MATIJEVIfDepartment of Chemistry, Clarkson College of Technology, Potsdam, New York
(Received July 8, 1960)
Values of intensity functions and total scattering coefficient are presented for n=2.105 and a=0. 2 (0.4)5.8; 6.0 (0.2) 15.0. The applicability of Penndorf's approximate methods to the total scattering coefficientis discussed.
WE have computed Mien scattering functions for'~v refractive index of 2.105 in connection with alight-scattering study of silver chloride aerosols. Inaddition to their usefulness in studying this particularexperimental system, these computations make avail-able a large number of results for a refractive index in arange where there have been very few data. This willnow permit a more careful analysis of the influence ofrefractive index upon the scattering functions and alsothe testing of existing interpolation procedures as wellas the development of improved ones.
The results are given in Table I. The size parametera is equal to 2 rr/X, where r is the particle radius and Xthe wavelength of radiation in the medium. The angleof observation y is the angle between the scatteringdirection and the reversed direction of the incidentbeam; hence 0° is the backward direction.
The angular distribution functions or intensity func-tions, i and i 2 , are proportional to the intensity of theperpendicular and horizontal components of thescattered light. For detailed definitions, the reader isreferred to the standard literature.- The total scatteringcoefficient K is the total flux scattered by a particlein all directions relative to the flux incident on thegeometrical cross section of the particle. The digitfollowing each number is the decimal exponent, e.g.,1.8099 -5 means 1.8099X10-1.
All computations were carried out with aid of theIBM 704 computer. For values of a less than 6, theLegendre functions were read into the computer fromthe tabulations of Gumprecht and Sliepcevich7 and, inthe cases for -y=450 and 1350, from our previous
*Supported in part by contract with the U. S. Army ChemicalCorps.
I G. Mie, Ann. Physik 25, 377 (1908).2 E. Matijevi6, M. Kerker, and K. F. Schulz, Discussions
Faraday Soc. (September, 1960).3 R. 0. Gumprecht and C. M. Sliepcevich, Light Scattering
Functions for Spherical Particles (University of Michigan Press,Ann Arbor, Michigan, 1951).
4 W. J. Pangonis, W. Heller, and A. Jacobson, Tables of LightScattering Functions for Spherical Particles (Wayne State Univer-sity Press, Detroit, Michigan, 1957).
5A. N. Lowan, Tables of Scattering Functions for SphericalParticles, Natl. Bur. Standards, (U. S.), Appl. Math. Series 4,Washington, Government Printing Office (1948).
6 H. C. Van de Hulst, Light Scattering l1a Small Particles (JohnWiley & Sons, Inc., New York, 1957).
R K. 0. Gumprecht and C. M. Sliepcevich, Functions of PartialDerivatives of Legendre Polynomials (University of MichiganPress, Ann Arbor, Michigan, 1951).
calculations.' The necessary Bessel functions werecomputed internally with the aid of the usual recursionrelations.9 This Bessel function routine failed for largervalues of a (a> 6) when higher-order functions werecalled for because of the loss of accuracy upon con-tinued iteration of the recursion relation.
For a> 6, a program developed by the NationalBureau of Standards was utilized. In order to maintainthe necessary precision in the calculation of the Besselfunctions, a double-precision interpretive system wasused (20 decimal places). In this case the Legendrefunctions were also computed internally using recursionrelations.
Despite the interest in scattering phenomena in suchdiverse areas as chemistry, meteorology, astrophysics,space science, microwave physics, etc., the number ofavailable scattering functions is still much too smallto cover the whole range of useful applications. All thesefunctions contain the two parameters, n and a. Van deHulst 0 has developed a lucid presentation of scatteringfunctions in terms of an m-a plane such that to eachpoint in the plane there exists a particular set offunctions corresponding to a particular scatteringsystem. The problem has been that not only is the regionof the m- a plane where exact Mie theory computationsare required quite extensive, but because of the fluc-tuating character of the functions, an extremely highdensity of points is necessary. This, in turn, has led to aneed for accurate interpolation schemes in order toobtain values for points in the plane intermediatebetween those available by exact computation.
Penndorf"'1 2 has recently collected together andstudied the existing values of the total scatteringcoefficient K for real values of the refractive index.His purpose was to develop numerical formulas whichwould permit rapid calculation of approximate valuesof K for intermediate refractive indexes. Since a largenumber of computations were available for refractive
8 M. Kerker and E. Matijevi6, J. Opt. Soc. Am. 50, 722 (1960).9 G. N. Watson, Theory of Bessel Functions (Cambridge Univer-
sity Press, Cambridge, Massachusetts, 1948).10 H. C. Van de Hulst, Optics of Spherical Particles (Recherches
Astronomiques de l'Observatoire d'Utrecht, XI part 1, Amsterdam,1946).
11 R. Penndorf, New Tables of Mie Scattering Functions forSpherical Particles (Geophysical Research Papers No. 45, Part 6,Air Force Cambridge Research Center, Bedford, Massachusetts,March, 1956).
12 R. Penndorf, J. Phys. Chem. 62, 1537 (1958).
87
VOLUME 51, NUMBER JANUARY, 1961
88 M. KERKER AND E. MATIJEVIC Vol 51
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indexes between 1.33 and 1.50 he has been mostsuccessful in this region.
The K values undergo major oscillations withincreasing a and upon these are superimposed a finestructure of ripples. Penndorf's formulas permitcalculation of a values at which the extrema of theoscillations occur as well as the values of K itself atthese points. Through these extrema a smooth curve isthen drawn, the fine structure being neglected. For therefractive index range 1.33 to 1.50, K values within 3%of the correct values are obtained by this procedure.
He used these to develop similar approximate relations,and the computations presented here -for m= 2.105offer the possibility of testing these relations.
In Fig. 1 we have plotted K as a function of a. Thedashed curve was obtained from Penndorf's formulas.The smooth curve is drawn through our computedpoints. It is quite obvious that these points would haveto be obtained even closer together in order to drawan accurate curve. We have hand computed the valuesat a= 1.94, 2.00, and 2.06 in order to delineate the maxi-ILU_ -L U- L.V'J.)
In view of the fact that extreme values of K only vary AlthU gr o a PEo- no.VV.
about 25% from the mean for most of the range Although Penndorf's method does locathe apositioncovered, this is not an insignificant error. of the major oscillations, the errors in the quantitative
Above m= 1.60, Penndorf had available only a values of K are too great to make them useful.limited number of scattering functions at m= 2.0.1513,14 Obviously, the fine structure cannot be neglected for
13 M. Kerker and H. Perlee, J. Opt. Soc. Am. 43, 49 (1953). this range of refractive index. Murley1" has drawn the14 M. Kerker and A. L. Cox, J. Opt. Soc. Am. 45, 1080 (1955),
Document 4677 American Documentation Institute. 15 R. D. Murley, J. Phys. Chem. 64, 161 (1960).
N2 . KERIX<R AND I,'-. MIATIJRVI(
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F;IG. 1. Total scattering function, K vs a for m1=2.105. Thesquares connected by the dashed line represent extrema computedby Penndorf's approximate method.
same conclusion based on a limited number of compu-tations for mn= 1.83 1.
Penndorf has also carried out a similar investigation
Vol. 51
for forward scattering (by= 180').16 In this case, hederives an approximate relation between i, i2, and Kwhich we give in the following form
i = i2 -O a'K' 16. - (1)
We have found that for m = 2.105, Eq. (1) is obeyedwith a maximum error of about 4% for a> 5. However,since i and i 2 at y= 1800 can be computed from thesame data as K itself with very little difficulty, theredoes not seem to be much advantage in this result.
The intensity functions at angles other than 0 and180° fluctuate much more erratically than K which isactually the integral of all of the intensity functions.It would seem to us, that the prospect for developinginterpolative formulas for these would be most formid-able. In view of these difficulties and with the availa-bility of high speed computers, the filling of the m-adomain with rigorously computed values is a morerealistic program than filling it with approximatevalues.
16 R. Penndorf, llie Scattering in the Forward Area (TechnicalReport RAD-TR-60-10, February, 1960, Air Force CambridgeResearch Center, Bedford, Massachusetts).
